A spectral based goodness-of-fit test for stochastic block models

Community detection in complex networks has attracted considerable attention, however, most existing methods need the number of communities to be specified beforehand. In this paper, a goodness-of-fit test based on the linear spectral statistic of the centered and rescaled adjacency matrix for the stochastic block model is proposed. We prove that the proposed test statistic converges in distribution to the standard Gaussian distribution under the null hypothesis. The proof uses some recent advances in generalized Wigner matrices. Simulations and real data examples show that our proposed test statistic performs well. This paper extends the work of Dong et al. [Information Science 512 (2020) 1360-1371].